// Magma code for working with number field 37.37.81381208133441979421709122744091225498491936628940230588748580298513087650630871328595025812353503688138712627681.1

// Some of these functions may take a long time to execute (this depends on the field).

// Define the number field: 
R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 - 666*x^35 - 481*x^34 + 193399*x^33 + 256521*x^32 - 32436827*x^31 - 60004084*x^30 + 3503452596*x^29 + 8113398850*x^28 - 257000630926*x^27 - 704292185114*x^26 + 13150853376643*x^25 + 41268101524916*x^24 - 474337598223255*x^23 - 1672482254287228*x^22 + 12032466243732809*x^21 + 47286637172470867*x^20 - 211502629431882231*x^19 - 929675982328753625*x^18 + 2498958523266151636*x^17 + 12536142236443615902*x^16 - 18791897342800992970*x^15 - 113239929214497210129*x^14 + 80927529107744817385*x^13 + 663966210336111429023*x^12 - 146546651897009713442*x^11 - 2439258160361411529393*x^10 - 154326326264692800324*x^9 + 5386184266991634344483*x^8 + 1055793633392402952768*x^7 - 6769150214799915444440*x^6 - 1552196159553019851407*x^5 + 4355671676506075568364*x^4 + 921754431147330625307*x^3 - 1108584774969076393499*x^2 - 217498966742521653325*x + 51140551819476687829);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

// Discriminant: 
OK := Integers(K); Discriminant(OK);

// Ramified primes: 
PrimeDivisors(Discriminant(OK));

// Autmorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Unit group: 
UK, fUK := UnitGroup(K);

// Unit rank: 
UnitRank(K);

// Generator for roots of unity: 
K!f(TU.1) where TU,f is TorsionUnitGroup(K);

// Fundamental units: 
[K|fUK(g): g in Generators(UK)];

// Regulator: 
Regulator(K);

// Analytic class number formula: 
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^37 - 666*x^35 - 481*x^34 + 193399*x^33 + 256521*x^32 - 32436827*x^31 - 60004084*x^30 + 3503452596*x^29 + 8113398850*x^28 - 257000630926*x^27 - 704292185114*x^26 + 13150853376643*x^25 + 41268101524916*x^24 - 474337598223255*x^23 - 1672482254287228*x^22 + 12032466243732809*x^21 + 47286637172470867*x^20 - 211502629431882231*x^19 - 929675982328753625*x^18 + 2498958523266151636*x^17 + 12536142236443615902*x^16 - 18791897342800992970*x^15 - 113239929214497210129*x^14 + 80927529107744817385*x^13 + 663966210336111429023*x^12 - 146546651897009713442*x^11 - 2439258160361411529393*x^10 - 154326326264692800324*x^9 + 5386184266991634344483*x^8 + 1055793633392402952768*x^7 - 6769150214799915444440*x^6 - 1552196159553019851407*x^5 + 4355671676506075568364*x^4 + 921754431147330625307*x^3 - 1108584774969076393499*x^2 - 217498966742521653325*x + 51140551819476687829);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

// Intermediate fields: 
L := Subfields(K); L[2..#L];

// Galois group: 
G = GaloisGroup(K);

// Frobenius cycle types: 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];